 2.4.1E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.2E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.3E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.4E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.5E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.6E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.7E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.8E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.9E: In Exercises 1–9, assume that the matrices are partitioned conforma...
 2.4.10E: The inverse of Find P, Q, and R.
 2.4.11E: In Exercises 11 and 12, mark each statement True or False. Justify ...
 2.4.12E: In Exercises 11 and 12, mark each statement True or False. Justify ...
 2.4.13E: Let where B and C are square. Show that A is invertible if and only...
 2.4.14E: Show that the block upper triangular matrix A in Example 5 is inver...
 2.4.15E: When a deep space probe is launched, corrections may be necessary t...
 2.4.16E: Let If A11 is invertible, then the matrix is called the Schur compl...
 2.4.17E: Suppose the block matrix A on the left side of (7) is invertible an...
 2.4.18E: Let X be an m × n data matrix such that XT X is invertible, and let...
 2.4.19E: In the study of engineering control of physical systems, a standard...
 2.4.20E: In the study of engineering control of physical systems, a standard...
 2.4.21E: a. Verify that b. Use partitioned matrices to show that M2 = I when
 2.4.22E: Generalize the idea of Exercise 21 by constructing a 6 × 6 matrix s...
 2.4.23E: Use partitioned matrices to prove by induction that the product of ...
 2.4.24E: Use partitioned matrices to prove by induction that for n = 2, 3,…,...
 2.4.25E: Without using row reduction, find the inverse of
 2.4.26E: [M] For block operations, it may be necessary to access or enter su...
 2.4.27E: [M] Suppose memory or size restrictions prevent a matrix program fr...
Solutions for Chapter 2.4: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 2.4
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.4 includes 27 full stepbystep solutions. Since 27 problems in chapter 2.4 have been answered, more than 30310 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.